Origin (ZeroSet)

ZeroSet{N} <: AbstractSingleton{N}

Type that represents the zero set, i.e., the set that only contains the origin.

Fields

• dim – the ambient dimension of this zero set

dim(Z::ZeroSet)

Return the ambient dimension of this zero set.

Input

• Z – a zero set, i.e., a set that only contains the origin

Output

The ambient dimension of the zero set.

σ(d::AbstractVector, Z::ZeroSet)

Return the support vector of a zero set.

Input

• d – direction
• Z – a zero set, i.e., a set that only contains the origin

Output

The returned value is the origin since it is the only point that belongs to this set.

ρ(d::AbstractVector, Z::ZeroSet)

Evaluate the support function of a zero set in a given direction.

Input

• d – direction
• Z – a zero set, i.e., a set that only contains the origin

Output

0.

∈(x::AbstractVector, Z::ZeroSet)

Check whether a given point is contained in a zero set.

Input

• x – point/vector
• Z – zero set

Output

true iff $x ∈ Z$.

Examples

julia> Z = ZeroSet(2);

julia> [1.0, 0.0] ∈ Z
false
julia> [0.0, 0.0] ∈ Z
true

rand(::Type{ZeroSet}; [N]::Type{<:Real}=Float64, [dim]::Int=2,
     [rng]::AbstractRNG=GLOBAL_RNG, [seed]::Union{Int, Nothing}=nothing)

Create a zero set (note that there is nothing to randomize).

Input

• ZeroSet – type for dispatch
• N – (optional, default: Float64) numeric type
• dim – (optional, default: 2) dimension
• rng – (optional, default: GLOBAL_RNG) random number generator
• seed – (optional, default: nothing) seed for reseeding

Output

The (only) zero set of the given numeric type and dimension.

element(S::ZeroSet{N}) where {N}

Return the element of a zero set.

Input

• S – zero set

Output

The element of the zero set, i.e., a zero vector.

element(S::ZeroSet{N}, ::Int) where {N}

Return the i-th entry of the element of a zero set.

Input

• S – zero set
• i – dimension

Output

The i-th entry of the element of the zero set, i.e., 0.

linear_map(M::AbstractMatrix, Z::ZeroSet)

Concrete linear map of a zero set.

Input

• M – matrix
• Z – zero set

Output

The zero set whose dimension matches the output dimension of the given matrix.

translate(Z::ZeroSet, v::AbstractVector)

Translate (i.e., shift) a zero set by a given vector.

Input

• Z – zero set
• v – translation vector

Output

A singleton containing the vector v.

center(Z::ZeroSet{N}, i::Int) where {N}

Return the center along a given dimension of a zero set.

Input

• Z – zero set
• i – dimension of interest

Output

The center along a given dimension of the zero set.

rectify(Z::ZeroSet)

Concrete rectification of a zero set.

Input

• Z – zero set

Output

The same set.